Consider the energy cycle depicted in the P-V diagram below with 2 stages at constant volume and 2 stages at constant pressure

Initial state 1 – a fixed mass of gas volume V pressure P and temperature T_{1}

Stage 1-2 the gas is heated at constant volume to
temperature T_{2}

raising pressure to P′
Heat input C_{v} (T_{2} - T_{1})

Stage 2-3 the gas is heated at constant pressure to
temperature T_{3}

and volume V′
Heat input C_{p} (T_{3} - T_{2})

Stage 3-4 work done by gas at constant volume so that pressure

falls to P and temperature fall to T_{4}

There is no heat input Q = DU + W = 0

work done by the gas C_{v} (T_{3} – T_{4})

Stage 4-1 heat taken out of the gas into energy sink at constant

pressure so that volume returns to V and temperature to T_{1}

Efficiency = __useful work
achieved__

total heat input

=
C_{v} (T_{3} - T_{4})

__ __

C_{v} (T_{2} - T_{1}) + C_{p}(T_{3} - T_{2})

Now consider the energy cycle portrayed approximating towards a constant volume heat engine

As V′ ®
V T_{4} ®
T_{1 }T_{3} ®
T_{2 }and (T_{3} – T_{2})
® 0

Efficiency ®
C_{v} (T_{2} - T_{1}) ®
100%

__ __

C_{v} (T_{2} - T_{1})

If an energy cycle can be devised with the entire heat input and extraction of work at constant volume it will have a maximum theoretical efficiency of 100%.